On positive solutions of a nonlocal fractional boundary value problem

In this paper, we investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation. Firstly, we give Green’s function and prove its positivity; secondly, the uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions; thirdly, by means of the fixed point index theory, we obtain some existence results of positive solution. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.     

Existence of periodic solutions for a system of delay differential equations

In this paper we mainly study the existence of periodic solutions for a system of delay differential equations representing a simple two-neuron network model of Hopfield type with time-delayed connections between the neurons. We first examine the local stability of the trivial solution, propose some sufficient conditions for the uniqueness of equilibria and then apply the Poincaré-Bendixson theorem for monotone cyclic feedback delayed systems to establish the existence of periodic solutions. In addition, a sufficient condition that ensures the trivial solution to be globally exponentially stable is also given. Numerical examples are provided to support the theoretical analysis.     

Existence and uniqueness of solutions of piecewise nonlinear systems

In this paper, we consider the existence and the uniqueness of solutions of piecewise nonlinear systems. We will present some necessary and sufficient conditions for the existence and the uniqueness of solutions of this class of systems.     

Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative by using the monotone iterative method.     

Rate of decay of stable periodic solutions of Duffing equations

In this paper, we consider the second-order equations of Duffing type. Bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution. Furthermore, the stability of the unique periodic solution is analyzed; the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution.     

Almost periodic solutions for abstract impulsive differential equations

This paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space. The investigations are carried out by means of the fractional powers of operators. We construct an example to illustrate the feasibility of our results.     

Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems

In this paper we establish the existence and the uniqueness of positive solutions for Dirichlet boundary value problems of nonlinear elliptic equations with singularity. We obtain the existence and the uniqueness by using the mixed monotone method in the cone theory. Moreover, we give an iterative method of constructing the solution. The rate of convergence of the iterative sequence is analyzed.     

Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order

This article introduces a new concept of upper and lower solutions and studies the existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order. The sufficient conditions for the existence and uniqueness of solutions are obtained by using the monotone iterative method, meantime, the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution are given. Some results of previous literature are extended and improved. A numerical example is also included to illustrate the effectiveness of the proposed results.     

Entire solutions in periodic lattice dynamical systems

This paper deals with entire solutions of periodic lattice dynamical systems. Unlike homogeneous problems, the periodic equation studied here lacks symmetry between increasing and decreasing pulsating traveling fronts, which affects the construction of entire solutions. In the bistable case, the existence, uniqueness and Liapunov stability of entire solutions are proved by constructing different sub- and supersolutions. In the monostable case, the existence and asymptotic behavior of spatially periodic solutions connecting two steady states are first established. Some new types of entire solutions are then constructed by combining leftward and rightward pulsating traveling fronts with different speeds and a spatially periodic solution. Various qualitative features of the entire solutions are also investigated.     

Monotone iterative technique for functional differential equations with retardation and anticipation

The method of the monotone iterative technique together with coupled lower and upper solutions is employed to prove existence and uniqueness results for functional differential equations involving retardation and anticipation.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

The generalized quasilinearization method for second-order three-point boundary value problems

In this paper, we obtain the existence and uniqueness results for general second-order three-point boundary value problems by applying the method of upper and lower solutions together with Leray–Schauder degree, as well as obtaining monotone iteration schemes which converge quadratically to the unique solution of some specific second-order three-point boundary value problem by using the method of quasilinearization.     

Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions

This paper describes the method of quasilinearization for first-order nonlinear impulsive functional differential equations with anti-periodic boundary conditions. A monotone iterative technique coupled with lower and upper solutions is employed to obtain sequences of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand.     

二阶三点边值问题解的存在性与唯一性

利用单调迭代方法,研究了反序上、下解条件下的二阶微分方程三点边值问题解的存在性与唯一性,分别得到解存在与唯一的充分条件,在满足解的唯一性的条件下,给出了求解的迭代序列及误差估计式.     

Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems

This paper investigates the existence of positive solutions of singular multi-point boundary value problems of fourth order ordinary differential equation with p-Laplacian. A necessary and sufficient condition for the existence of C²[0,1] positive solution as well as pseudo-C³[0,1] positive solution is given by means of the fixed point theorems on cones.     

Positive solutions for a nonlinear discrete fractional boundary value problem with a $p$-Laplacian operator

In this paper using the monotone iterative technique we establish the existence and uniqueness of positive solutions for a nonlinear discrete fractional boundary value problem with a $p$-Laplacian operator. Also we discuss an iterative sequence which yields the approximate solution for this problem.     

Necessary and sufficient condition for the existence of positive solutions of a coupled system for elastic beam equations

In this work, by using fixed point theorem and monotone iterative technique, the author establishes a necessary and sufficient condition of the existence of positive solutions for a class of nonlinear singular elastic beam differential system under reasonable conditions. Moreover, some uniqueness results of positive solution and the estimate of convergent rate of the iterative sequence of solution are obtained.     

Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system

In this paper, we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear Hamiltonian systems with impulsive conditions via variational methods. However, without impulses, there is no homoclinic or periodic solution for the system considered in this paper. Moreover, our results can be used to study the existence of periodic and homoclinic solutions of difference equations.     

Periodic solutions of functional-differential equations of point type

We study periodic solutions of a functional-differential equation of point type. We state conditions for the existence and uniqueness of an ω-periodic solution of the original nonlinear functional-differential equation of point type. An iterative process for constructing such a solution is described, and its convergence rate is estimated.